Tuesday, April 01, 2014

The Pythagoreans were a secret society in ancient Greece, a mystical brotherhood dedicated to mathematical study. Their motto was "All is number." Esoteric groups such as the Freemasons and Rosicrucians claim descent from the Pythagorean Brotherhood.

The central assumption of the Pythagorean brotherhood was that the entire universe could be explained by careful application of whole numbers or their ratios (i.e. fractions, or rational numbers). Legend has it that when a member of the cult of Pythagoras proved that even something as simple as the diagonal of a square could not be measured using whole numbers or their ratios, he was summarily executed to avoid insulting prevailing Pythagorean doctrine.

Nevertheless, Pythagorean theories have been applied over the centuries to such diverse problems such as building the great pyramids of Egypt, establishing musical scales, and mapping the orbits of planets.

The good ol' Pythagorean Theorem is one of the most fundamental formulas in all of mathematics. Falsely attributed to Pythagoras of Samos, this famous rule defines the relationship between the lengths of the sides of all right triangles:

a^2 + b^2 = c^2.

Pythagorean Triples are trios of whole numbers that conform to the requirements of the Pythagorean Theorem, and therefore measure the three sides of a right triangle: the two "legs" forming the right angle, and the "hypotenuse" (not "hippopotamus") opposite the right angle.

Here's a fun and simple way to generate Pythagorean Triples from any given natural number (indeed, all Pythagorean Triples can be generated this way):

1. Pick a natural number larger than two. Call this number n.

2. If n is even: let y be 1 and let x be half of n; otherwise (if n is odd) first subtract 1 from n, then let y be half of this new number and let x be one more than y.

3. The following equations will generate Pythagorean Triples for any such x, y:

About

I help students to learn and love mathematics, produce outstanding results on standardized tests, and thrive as productive, successful scholars. Students of mine go from D's and F's to A's and B's, raise their SAT scores by as much as 450 points or more, and dramatically increase their confidence and competence as thinkers and problem solvers. I began my career as a private teacher giving music lessons at the age of 16, and have operated a professional academic coaching practice for the past four decades. Since 1977, it's been my privilege to serve as academic coach, tutor, and mentor to well over 1500 private students of all ages, helping them to reach the pinnacle of their abilities, attain ambitious educational goals, and achieve enduring success as students and young people. My interests include: Bollywood films, art, archery, Macs, music, recreational math and science, philosophy, psychology, history, comedy, saving the world, and visiting the in-laws in Bali, Indonesia.

Quotes

-

The secret of getting ahead is getting started.

I have never let my schooling interfere with my education.

As to the militant suffragettes, I have noted that many women believe in militant methods. You might advocate one way of securing the rights and I might advocate another, they both might help to bring about the result desired. To win freedom always involves hard fighting. I believe in women doing what they deem necessary to secure their rights.

Travel is fatal to prejudice, bigotry, and narrow-mindedness.

A successful book is not made of what is in it, but of what is left out of it.

Marriage—yes, it is the supreme felicity of life. I concede it. And it is also the supreme tragedy of life. The deeper the love the surer the tragedy. And the more disconsolating when it comes.

It's not the size of the dog in the fight, it's the size of the fight in the dog.

“Substitute 'damn' every time you're inclined to write 'very;' your editor will delete it and the writing will be just as it should be.”

“The fear of death follows from the fear of life. A man who lives fully is prepared to die at any time.”

Mark Twain

-

"If you cannot solve the proposed problem, try to solve first a simpler related problem."

"Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize."

"A mathematics teacher is a midwife to ideas."

"A GREAT discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery."

“Whatever the subject, what the teacher really teaches is himself.”

George Pólya

-

"The good life is one inspired by love and guided by knowledge."

"No nation was ever so virtuous as each believes itself, and none was ever so wicked as each believes the other."

"The desire to understand the world and the desire to reform it are the two great engines of progress."

"Although this may seem a paradox, all exact science is dominated by the idea of approximation."

"Boredom is a vital problem for the moralist, since at least half the sins of mankind are caused by the fear of it."

Betrand Russell

-

"Everything should be made as simple as possible, but no simpler."

"All religions, arts and sciences are branches of the same tree. All these aspirations are directed toward ennobling man's life, lifting it from the sphere of mere physical existence and leading the individual towards freedom."

"Imagination is more important than knowledge."

"Do not worry about your difficulties in mathematics, I assure you that mine are greater."

"There are two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle."

Albert Einstein

-

"Teachers have always been—and let us hope, always will be—in the business of trying to get the soul out of bed, out of her deep, habitual sleep."

"Money often costs too much."

"Sometimes a scream is better than a thesis."

"There is properly no history; only biography."

"Standing on the bare ground, — my head bathed by the blithe air, and uplifted into infinite space, — all mean egotism vanishes. I become a transparent eye-ball; I am nothing; I see all; the currents of the Universal Being circulate through me; I am part or particle of God."